This interdisciplinary research project combines methods and motivating questions arising in theoretical physics with mathematical methods of number theory and arithmetic geometry. Surprising connections between these two very different areas of research have been observed in recent years: the principal investigator has been involved in the study of these relationships between high-energy physics and number theory, and she is currently developing new mathematical approaches towards a better understanding of this mysterious connection. The goal of this project is to extend this connection beyond its original occurrence in quantum field theory to other areas, including statistical physics, string theory, and quantum information.
This project comprises several related investigations. The Feynman diagram calculations used in high-energy physics to describe events in particle physics experiments have a rich and not yet fully understood mathematical structure, involving motives and periods of algebraic varieties, which are objects of study in algebraic geometry, arithmetic geometry, and number theory. This project aims to develop new techniques to improve understanding of these arithmetic structures in quantum field theory. Similar mathematical structures based on periods and motives of algebraic varieties occur in models of modified gravity developed for applications to cosmology; this new, unexpected development will be investigated in depth as part of the research. The project will also study a framework for a discretization of the "holographic correspondence" of string theory, between gravity on a bulk space and conformal field theory on the boundary, built in number-theoretic terms using p-adic numbers. The project aims to develop this p-adic form of the AdS/CFT correspondence into a larger theoretical framework, with a view to explaining in arithmetic terms the emergence of spacetime geometry from information. The work will also continue development of another novel approach to the relation between geometry and information, a construction of anyon models of topological quantum computation based on special solutions (gravitational instantons) of the Einstein equations, with singularities along two-dimensional surfaces. The investigator is also pursuing new number theoretic approaches to quantum statistical mechanics, where the properties of the quantum systems correspond to properties of certain classes of zeta functions studied in analytic number theory, and the equilibrium states of the systems capture Galois symmetries. Finally, the investigator is developing a new approach to distributed computing in theoretical computer science, based on the technique of dynamical triangulations developed in physics as a model of quantum gravity.